A sum of money invested at compound interest amounts to ? 800 in 2 yr and ? 840 in 3 yr. Find the rate of interest per annum and the sum.

Answer

To solve the problem, we need to find the principal amount (the sum of money invested) and the rate of interest per annum based on the information provided about the amounts after 2 years and 3 years.

### Step 1: Understand the given information
We know:
- Amount after 2 years (A2) = ₹800
- Amount after 3 years (A3) = ₹840

### Step 2: Use the formula for compound interest
The amount after n years can be calculated using the formula:
\[ A = P(1 + r)^n \]
Where:
- \( A \) = Amount after n years
- \( P \) = Principal amount (initial sum)
- \( r \) = Rate of interest per annum (in decimal)
- \( n \) = Number of years

### Step 3: Set up equations based on the given amounts
From the information given:
1. For 2 years:
\[ A2 = P(1 + r)^2 = 800 \] (Equation 1)

2. For 3 years:
\[ A3 = P(1 + r)^3 = 840 \] (Equation 2)

### Step 4: Divide Equation 2 by Equation 1
To eliminate \( P \), we can divide the second equation by the first equation:
\[ \frac{A3}{A2} = \frac{P(1 + r)^3}{P(1 + r)^2} \]
This simplifies to:
\[ \frac{840}{800} = (1 + r) \]
Calculating the left side:
\[ 1.05 = (1 + r) \]

### Step 5: Solve for \( r \)
Now, we can find \( r \):
\[ r = 1.05 - 1 = 0.05 \]
Converting \( r \) to percentage:
\[ r = 0.05 \times 100 = 5\% \]

### Step 6: Substitute \( r \) back to find \( P \)
Now that we have \( r \), we can substitute it back into either Equation 1 or Equation 2 to find \( P \). Let's use Equation 1:
\[ 800 = P(1 + 0.05)^2 \]
Calculating \( (1 + 0.05)^2 \):
\[ 800 = P(1.1025) \]
Now, solving for \( P \):
\[ P = \frac{800}{1.1025} \approx 726.19 \]

### Summary of Results
- Principal amount (P) ≈ ₹726.19
- Rate of interest (r) = 5% per annum